/* mpfr_sub1sp -- internal function to perform a "real" substraction All the op must have the same precision Copyright 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 Free Software Foundation, Inc. Contributed by the AriC and Caramel projects, INRIA. This file is part of the GNU MPFR Library. The GNU MPFR Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. The GNU MPFR Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ #define MPFR_NEED_LONGLONG_H #include "mpfr-impl.h" /* Check if we have to check the result of mpfr_sub1sp with mpfr_sub1 */ #ifdef WANT_ASSERT # if WANT_ASSERT >= 2 int mpfr_sub1sp2 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode); int mpfr_sub1sp (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { mpfr_t tmpa, tmpb, tmpc; int inexb, inexc, inexact, inexact2; mpfr_init2 (tmpa, MPFR_PREC (a)); mpfr_init2 (tmpb, MPFR_PREC (b)); mpfr_init2 (tmpc, MPFR_PREC (c)); inexb = mpfr_set (tmpb, b, MPFR_RNDN); MPFR_ASSERTN (inexb == 0); inexc = mpfr_set (tmpc, c, MPFR_RNDN); MPFR_ASSERTN (inexc == 0); inexact2 = mpfr_sub1 (tmpa, tmpb, tmpc, rnd_mode); inexact = mpfr_sub1sp2(a, b, c, rnd_mode); if (mpfr_cmp (tmpa, a) || inexact != inexact2) { fprintf (stderr, "sub1 & sub1sp return different values for %s\n" "Prec_a = %lu, Prec_b = %lu, Prec_c = %lu\nB = ", mpfr_print_rnd_mode (rnd_mode), (unsigned long) MPFR_PREC (a), (unsigned long) MPFR_PREC (b), (unsigned long) MPFR_PREC (c)); mpfr_fprint_binary (stderr, tmpb); fprintf (stderr, "\nC = "); mpfr_fprint_binary (stderr, tmpc); fprintf (stderr, "\nSub1 : "); mpfr_fprint_binary (stderr, tmpa); fprintf (stderr, "\nSub1sp: "); mpfr_fprint_binary (stderr, a); fprintf (stderr, "\nInexact sp = %d | Inexact = %d\n", inexact, inexact2); MPFR_ASSERTN (0); } mpfr_clears (tmpa, tmpb, tmpc, (mpfr_ptr) 0); return inexact; } # define mpfr_sub1sp mpfr_sub1sp2 # endif #endif /* Debugging support */ #ifdef DEBUG # undef DEBUG # define DEBUG(x) (x) #else # define DEBUG(x) /**/ #endif /* Rounding Sub */ /* compute sgn(b)*(|b| - |c|) if |b|>|c| else -sgn(b)*(|c| -|b|) Returns 0 iff result is exact, a negative value when the result is less than the exact value, a positive value otherwise. */ /* A0...Ap-1 * Cp Cp+1 .... * <- C'p+1 -> * Cp = -1 if calculated from c mantissa * Cp = 0 if 0 from a or c * Cp = 1 if calculated from a. * C'p+1 = First bit not null or 0 if there isn't one * * Can't have Cp=-1 and C'p+1=1*/ /* RND = MPFR_RNDZ: * + if Cp=0 and C'p+1=0,1, Truncate. * + if Cp=0 and C'p+1=-1, SubOneUlp * + if Cp=-1, SubOneUlp * + if Cp=1, AddOneUlp * RND = MPFR_RNDA (Away) * + if Cp=0 and C'p+1=0,-1, Truncate * + if Cp=0 and C'p+1=1, AddOneUlp * + if Cp=1, AddOneUlp * + if Cp=-1, Truncate * RND = MPFR_RNDN * + if Cp=0, Truncate * + if Cp=1 and C'p+1=1, AddOneUlp * + if Cp=1 and C'p+1=-1, Truncate * + if Cp=1 and C'p+1=0, Truncate if Ap-1=0, AddOneUlp else * + if Cp=-1 and C'p+1=-1, SubOneUlp * + if Cp=-1 and C'p+1=0, Truncate if Ap-1=0, SubOneUlp else * * If AddOneUlp: * If carry, then it is 11111111111 + 1 = 10000000000000 * ap[n-1]=MPFR_HIGHT_BIT * If SubOneUlp: * If we lose one bit, it is 1000000000 - 1 = 0111111111111 * Then shift, and put as last bit x which is calculated * according Cp, Cp-1 and rnd_mode. * If Truncate, * If it is a power of 2, * we may have to suboneulp in some special cases. * * To simplify, we don't use Cp = 1. * */ int mpfr_sub1sp (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { mpfr_exp_t bx,cx; mpfr_uexp_t d; mpfr_prec_t p, sh, cnt; mp_size_t n; mp_limb_t *ap, *bp, *cp; mp_limb_t limb; int inexact; mp_limb_t bcp,bcp1; /* Cp and C'p+1 */ mp_limb_t bbcp = (mp_limb_t) -1, bbcp1 = (mp_limb_t) -1; /* Cp+1 and C'p+2, gcc claims that they might be used uninitialized. We fill them with invalid values, which should produce a failure if so. See README.dev file. */ MPFR_TMP_DECL(marker); MPFR_TMP_MARK(marker); MPFR_ASSERTD(MPFR_PREC(a) == MPFR_PREC(b) && MPFR_PREC(b) == MPFR_PREC(c)); MPFR_ASSERTD(MPFR_IS_PURE_FP(b)); MPFR_ASSERTD(MPFR_IS_PURE_FP(c)); /* Read prec and num of limbs */ p = MPFR_PREC(b); n = (p-1)/GMP_NUMB_BITS+1; /* Fast cmp of |b| and |c|*/ bx = MPFR_GET_EXP (b); cx = MPFR_GET_EXP (c); if (MPFR_UNLIKELY(bx == cx)) { mp_size_t k = n - 1; /* Check mantissa since exponent are equals */ bp = MPFR_MANT(b); cp = MPFR_MANT(c); while (k>=0 && MPFR_UNLIKELY(bp[k] == cp[k])) k--; if (MPFR_UNLIKELY(k < 0)) /* b == c ! */ { /* Return exact number 0 */ if (rnd_mode == MPFR_RNDD) MPFR_SET_NEG(a); else MPFR_SET_POS(a); MPFR_SET_ZERO(a); MPFR_RET(0); } else if (bp[k] > cp[k]) goto BGreater; else { MPFR_ASSERTD(bp[k] c */ BGreater: MPFR_SET_SAME_SIGN(a,b); } /* Now b > c */ MPFR_ASSERTD(bx >= cx); d = (mpfr_uexp_t) bx - cx; DEBUG (printf ("New with diff=%lu\n", (unsigned long) d)); if (MPFR_UNLIKELY(d <= 1)) { if (MPFR_LIKELY(d < 1)) { /* <-- b --> <-- c --> : exact sub */ ap = MPFR_MANT(a); mpn_sub_n (ap, MPFR_MANT(b), MPFR_MANT(c), n); /* Normalize */ ExactNormalize: limb = ap[n-1]; if (MPFR_LIKELY(limb)) { /* First limb is not zero. */ count_leading_zeros(cnt, limb); /* cnt could be == 0 <= SubD1Lose */ if (MPFR_LIKELY(cnt)) { mpn_lshift(ap, ap, n, cnt); /* Normalize number */ bx -= cnt; /* Update final expo */ } /* Last limb should be ok */ MPFR_ASSERTD(!(ap[0] & MPFR_LIMB_MASK((unsigned int) (-p) % GMP_NUMB_BITS))); } else { /* First limb is zero */ mp_size_t k = n-1, len; /* Find the first limb not equal to zero. FIXME:It is assume it exists (since |b| > |c| and same prec)*/ do { MPFR_ASSERTD( k > 0 ); limb = ap[--k]; } while (limb == 0); MPFR_ASSERTD(limb != 0); count_leading_zeros(cnt, limb); k++; len = n - k; /* Number of last limb */ MPFR_ASSERTD(k >= 0); if (MPFR_LIKELY(cnt)) mpn_lshift(ap+len, ap, k, cnt); /* Normalize the High Limb*/ else { /* Must use DECR since src and dest may overlap & dest>=src*/ MPN_COPY_DECR(ap+len, ap, k); } MPN_ZERO(ap, len); /* Zeroing the last limbs */ bx -= cnt + len*GMP_NUMB_BITS; /* Update Expo */ /* Last limb should be ok */ MPFR_ASSERTD(!(ap[len]&MPFR_LIMB_MASK((unsigned int) (-p) % GMP_NUMB_BITS))); } /* Check expo underflow */ if (MPFR_UNLIKELY(bx < __gmpfr_emin)) { MPFR_TMP_FREE(marker); /* inexact=0 */ DEBUG( printf("(D==0 Underflow)\n") ); if (rnd_mode == MPFR_RNDN && (bx < __gmpfr_emin - 1 || (/*inexact >= 0 &&*/ mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } MPFR_SET_EXP (a, bx); /* No rounding is necessary since the result is exact */ MPFR_ASSERTD(ap[n-1] > ~ap[n-1]); MPFR_TMP_FREE(marker); return 0; } else /* if (d == 1) */ { /* | <-- b --> | <-- c --> */ mp_limb_t c0, mask; mp_size_t k; MPFR_UNSIGNED_MINUS_MODULO(sh, p); /* If we lose at least one bit, compute 2*b-c (Exact) * else compute b-c/2 */ bp = MPFR_MANT(b); cp = MPFR_MANT(c); k = n-1; limb = bp[k] - cp[k]/2; if (limb > MPFR_LIMB_HIGHBIT) { /* We can't lose precision: compute b-c/2 */ /* Shift c in the allocated temporary block */ SubD1NoLose: c0 = cp[0] & (MPFR_LIMB_ONE< ~ap[n-1]); /* No rounding is necessary since the result is exact */ MPFR_TMP_FREE(marker); return 0; } ap = MPFR_MANT(a); mask = ~MPFR_LIMB_MASK(sh); cp[0] &= mask; /* Delete last bit of c */ mpn_sub_n (ap, bp, cp, n); MPFR_SET_EXP(a, bx); /* No expo overflow! */ MPFR_ASSERTD( !(ap[0] & ~mask) ); /* Check last bits */ /* No normalize is needed */ MPFR_ASSERTD(ap[n-1] > ~ap[n-1]); /* Rounding is necessary since c0 = 1*/ /* Cp =-1 and C'p+1=0 */ bcp = 1; bcp1 = 0; if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { /* Even Rule apply: Check Ap-1 */ if (MPFR_LIKELY( (ap[0] & (MPFR_LIMB_ONE< We lose at least one bit*/ /* If b[k]>c'[k] => We don't lose any bit */ /* If k==-1 => We don't lose any bit AND the result is 100000000000 0000000000 00000000000 */ mp_limb_t carry; do { carry = cp[k]&MPFR_LIMB_ONE; k--; } while (k>=0 && bp[k]==(carry=cp[k]/2+(carry<<(GMP_NUMB_BITS-1)))); if (MPFR_UNLIKELY(k<0)) { /*If carry then (sh==0 and Virtual c'[-1] > Virtual b[-1]) */ if (MPFR_UNLIKELY(carry)) /* carry = cp[0]&MPFR_LIMB_ONE */ { /* FIXME: Can be faster? */ MPFR_ASSERTD(sh == 0); goto SubD1Lose; } /* Result is a power of 2 */ ap = MPFR_MANT (a); MPN_ZERO (ap, n); ap[n-1] = MPFR_LIMB_HIGHBIT; MPFR_SET_EXP (a, bx); /* No expo overflow! */ /* No Normalize is needed*/ /* No Rounding is needed */ MPFR_TMP_FREE (marker); return 0; } /* carry = cp[k]/2+(cp[k-1]&1)<<(GMP_NUMB_BITS-1) = c'[k]*/ else if (bp[k] > carry) goto SubD1NoLose; else { MPFR_ASSERTD(bp[k]= p)) { ap = MPFR_MANT(a); MPFR_UNSIGNED_MINUS_MODULO(sh, p); /* We can't set A before since we use cp for rounding... */ /* Perform rounding: check if a=b or a=b-ulp(b) */ if (MPFR_UNLIKELY(d == p)) { /* cp == -1 and c'p+1 = ? */ bcp = 1; /* We need Cp+1 later for a very improbable case. */ bbcp = (MPFR_MANT(c)[n-1] & (MPFR_LIMB_ONE<<(GMP_NUMB_BITS-2))); /* We need also C'p+1 for an even more unprobable case... */ if (MPFR_LIKELY( bbcp )) bcp1 = 1; else { cp = MPFR_MANT(c); if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT)) { mp_size_t k = n-1; do { k--; } while (k>=0 && cp[k]==0); bcp1 = (k>=0); } else bcp1 = 1; } DEBUG( printf("(D=P) Cp=-1 Cp+1=%d C'p+1=%d \n", bbcp!=0, bcp1!=0) ); bp = MPFR_MANT (b); /* Even if src and dest overlap, it is ok using MPN_COPY */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { if (MPFR_UNLIKELY( bcp && bcp1==0 )) /* Cp=-1 and C'p+1=0: Even rule Apply! */ /* Check Ap-1 = Bp-1 */ if ((bp[0] & (MPFR_LIMB_ONE<P) Cp=%d Cp+1=%d C'p+1=%d\n", bcp!=0,bbcp!=0,bcp1!=0) ); /* Need to compute C'p+2 if d==p+1 and if rnd_mode=NEAREST (Because of a very improbable case) */ if (MPFR_UNLIKELY(d==p+1 && rnd_mode==MPFR_RNDN)) { cp = MPFR_MANT(c); if (MPFR_UNLIKELY(cp[n-1] == MPFR_LIMB_HIGHBIT)) { mp_size_t k = n-1; do { k--; } while (k>=0 && cp[k]==0); bbcp1 = (k>=0); } else bbcp1 = 1; DEBUG( printf("(D>P) C'p+2=%d\n", bbcp1!=0) ); } /* Copy mantissa B in A */ MPN_COPY(ap, MPFR_MANT(b), n); /* Round */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) goto truncate; MPFR_UPDATE_RND_MODE(rnd_mode, MPFR_IS_NEG(a)); if (rnd_mode == MPFR_RNDZ) goto sub_one_ulp; else /* rnd_mode = AWAY */ goto truncate; } } else { mpfr_uexp_t dm; mp_size_t m; mp_limb_t mask; /* General case: 2 <= d < p */ MPFR_UNSIGNED_MINUS_MODULO(sh, p); cp = MPFR_TMP_LIMBS_ALLOC (n); /* Shift c in temporary allocated place */ dm = d % GMP_NUMB_BITS; m = d / GMP_NUMB_BITS; if (MPFR_UNLIKELY(dm == 0)) { /* dm = 0 and m > 0: Just copy */ MPFR_ASSERTD(m!=0); MPN_COPY(cp, MPFR_MANT(c)+m, n-m); MPN_ZERO(cp+n-m, m); } else if (MPFR_LIKELY(m == 0)) { /* dm >=2 and m == 0: just shift */ MPFR_ASSERTD(dm >= 2); mpn_rshift(cp, MPFR_MANT(c), n, dm); } else { /* dm > 0 and m > 0: shift and zero */ mpn_rshift(cp, MPFR_MANT(c)+m, n-m, dm); MPN_ZERO(cp+n-m, m); } DEBUG( mpfr_print_mant_binary("Before", MPFR_MANT(c), p) ); DEBUG( mpfr_print_mant_binary("B= ", MPFR_MANT(b), p) ); DEBUG( mpfr_print_mant_binary("After ", cp, p) ); /* Compute bcp=Cp and bcp1=C'p+1 */ if (MPFR_LIKELY(sh)) { /* Try to compute them from C' rather than C (FIXME: Faster?) */ bcp = (cp[0] & (MPFR_LIMB_ONE<<(sh-1))) ; if (MPFR_LIKELY( cp[0] & MPFR_LIMB_MASK(sh-1) )) bcp1 = 1; else { /* We can't compute C'p+1 from C'. Compute it from C */ /* Start from bit x=p-d+sh in mantissa C (+sh since we have already looked sh bits in C'!) */ mpfr_prec_t x = p-d+sh-1; if (MPFR_LIKELY(x>p)) /* We are already looked at all the bits of c, so C'p+1 = 0*/ bcp1 = 0; else { mp_limb_t *tp = MPFR_MANT(c); mp_size_t kx = n-1 - (x / GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); DEBUG (printf ("(First) x=%lu Kx=%ld Sx=%lu\n", (unsigned long) x, (long) kx, (unsigned long) sx)); /* Looks at the last bits of limb kx (if sx=0 does nothing)*/ if (tp[kx] & MPFR_LIMB_MASK(sx)) bcp1 = 1; else { /*kx += (sx==0);*/ /*If sx==0, tp[kx] hasn't been checked*/ do { kx--; } while (kx>=0 && tp[kx]==0); bcp1 = (kx >= 0); } } } } else { /* Compute Cp and C'p+1 from C with sh=0 */ mp_limb_t *tp = MPFR_MANT(c); /* Start from bit x=p-d in mantissa C */ mpfr_prec_t x = p-d; mp_size_t kx = n-1 - (x / GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); MPFR_ASSERTD(p >= d); bcp = (tp[kx] & (MPFR_LIMB_ONE<=0 && tp[kx]==0); bcp1 = (kx>=0); } } DEBUG( printf("sh=%lu Cp=%d C'p+1=%d\n", sh, bcp!=0, bcp1!=0) ); /* Check if we can lose a bit, and if so compute Cp+1 and C'p+2 */ bp = MPFR_MANT(b); if (MPFR_UNLIKELY((bp[n-1]-cp[n-1]) <= MPFR_LIMB_HIGHBIT)) { /* We can lose a bit so we precompute Cp+1 and C'p+2 */ /* Test for trivial case: since C'p+1=0, Cp+1=0 and C'p+2 =0 */ if (MPFR_LIKELY(bcp1 == 0)) { bbcp = 0; bbcp1 = 0; } else /* bcp1 != 0 */ { /* We can lose a bit: compute Cp+1 and C'p+2 from mantissa C */ mp_limb_t *tp = MPFR_MANT(c); /* Start from bit x=(p+1)-d in mantissa C */ mpfr_prec_t x = p+1-d; mp_size_t kx = n-1 - (x/GMP_NUMB_BITS); mpfr_prec_t sx = GMP_NUMB_BITS-1-(x%GMP_NUMB_BITS); MPFR_ASSERTD(p > d); DEBUG (printf ("(pre) x=%lu Kx=%ld Sx=%lu\n", (unsigned long) x, (long) kx, (unsigned long) sx)); bbcp = (tp[kx] & (MPFR_LIMB_ONE<=0 && tp[kx]==0); bbcp1 = (kx>=0); DEBUG (printf ("(Pre) Scan done for %ld\n", (long) kx)); } } /*End of Bcp1 != 0*/ DEBUG( printf("(Pre) Cp+1=%d C'p+2=%d\n", bbcp!=0, bbcp1!=0) ); } /* End of "can lose a bit" */ /* Clean shifted C' */ mask = ~MPFR_LIMB_MASK (sh); cp[0] &= mask; /* Substract the mantissa c from b in a */ ap = MPFR_MANT(a); mpn_sub_n (ap, bp, cp, n); DEBUG( mpfr_print_mant_binary("Sub= ", ap, p) ); /* Normalize: we lose at max one bit*/ if (MPFR_UNLIKELY(MPFR_LIMB_MSB(ap[n-1]) == 0)) { /* High bit is not set and we have to fix it! */ /* Ap >= 010000xxx001 */ mpn_lshift(ap, ap, n, 1); /* Ap >= 100000xxx010 */ if (MPFR_UNLIKELY(bcp!=0)) /* Check if Cp = -1 */ /* Since Cp == -1, we have to substract one more */ { mpn_sub_1(ap, ap, n, MPFR_LIMB_ONE<= 10000xxx001 */ /* Final exponent -1 since we have shifted the mantissa */ bx--; /* Update bcp and bcp1 */ MPFR_ASSERTN(bbcp != (mp_limb_t) -1); MPFR_ASSERTN(bbcp1 != (mp_limb_t) -1); bcp = bbcp; bcp1 = bbcp1; /* We dont't have anymore a valid Cp+1! But since Ap >= 100000xxx001, the final sub can't unnormalize!*/ } MPFR_ASSERTD( !(ap[0] & ~mask) ); /* Rounding */ if (MPFR_LIKELY(rnd_mode == MPFR_RNDN)) { if (MPFR_LIKELY(bcp==0)) goto truncate; else if ((bcp1) || ((ap[0] & (MPFR_LIMB_ONE<=0 && ap[k]==0); if (MPFR_UNLIKELY(k<0)) { /* It is a power of 2! */ /* Compute Cp+1 if it isn't already compute (ie d==1) */ /* FIXME: Is this case possible? */ if (d == 1) bbcp=0; DEBUG( printf("(Truncate) Cp=%d, Cp+1=%d C'p+1=%d C'p+2=%d\n", \ bcp!=0, bbcp!=0, bcp1!=0, bbcp1!=0) ); MPFR_ASSERTN(bbcp != (mp_limb_t) -1); MPFR_ASSERTN((rnd_mode != MPFR_RNDN) || (bcp != 0) || (bbcp == 0) || (bbcp1 != (mp_limb_t) -1)); if (((rnd_mode != MPFR_RNDZ) && bcp) || ((rnd_mode == MPFR_RNDN) && (bcp == 0) && (bbcp) && (bbcp1))) { DEBUG( printf("(Truncate) Do sub\n") ); mpn_sub_1 (ap, ap, n, MPFR_LIMB_ONE << sh); mpn_lshift(ap, ap, n, 1); ap[0] |= MPFR_LIMB_ONE< MPFR_EXP(c) > emin if d == 1 : bx=MPFR_EXP(b). If we could lose any bits, the exact normalisation is called. if d >= p : bx=MPFR_EXP(b) >= MPFR_EXP(c) + p > emin After SubOneUlp, we could have one bit less. if 1 < d < p : bx >= MPFR_EXP(b)-2 >= MPFR_EXP(c) > emin if d == 1 : bx >= MPFR_EXP(b)-1 = MPFR_EXP(c) > emin. if d >= p : bx >= MPFR_EXP(b)-1 > emin since p>=2. */ MPFR_ASSERTD( bx >= __gmpfr_emin); /* if (MPFR_UNLIKELY(bx < __gmpfr_emin)) { DEBUG( printf("(Final Underflow)\n") ); if (rnd_mode == MPFR_RNDN && (bx < __gmpfr_emin - 1 || (inexact >= 0 && mpfr_powerof2_raw (a)))) rnd_mode = MPFR_RNDZ; MPFR_TMP_FREE(marker); return mpfr_underflow (a, rnd_mode, MPFR_SIGN(a)); } */ MPFR_SET_EXP (a, bx); MPFR_TMP_FREE(marker); MPFR_RET (inexact * MPFR_INT_SIGN (a)); }